Entropy for $k$-trees defined by $k$ transition matrices

In this work we consider Markov tree-shifts given by $k$ transition matrices, one for each of its $k$ directions. We analyse some topological properties introduced by arXiv:1509.01355 in order to answer some of the questions raised by those authors. Moreover, we provide a method to characterize the complexity function for Markov tree-shifts; this function is used to calculate the tree entropies defined by arXiv:1712.02251 and arXiv:1509.08325. We compare both entropies in order to determine some of its properties. Finally, the characterization of the complexity function is used to calculate the entropy of all binary Markov tree-shifts over the alphabet $\{0,1\}$.